The Braves said the former outfielder and pinch-hitting specialist died of congestive heart and lung failure.
Smith played in the big leagues for eight years, starting out with the Chicago Cubs, splitting a season between the Angels and Baltimore Orioles before joining the Braves for his final two seasons.
His son, Dwight Jr., has also spent time in majors, most recently with Baltimore in 2020
The elder Smith served mostly as a pinch-hitter with the Braves in 1995, hitting .252 with three homers and 21 RBI in 103 games.
Atlanta defeated Cleveland in six games to win the World Series, with Smith appearing three times as a pinch-hitter. He went 1 for 2 with a walk.